(a) Let be anti-self-adjoint, , on a complex inner-product space. Prove that every…

PHYSICS · Subjective · Fermi University Physics · Mathematical Methods for Physics · Linear algebra and vector spaces · Linear algebra and vector spaces

Conditions of Validity, Ch The spectral theorem, degeneracy and the matrices that will not diagonalise

Problem

(a) Let

T

be anti-self-adjoint,

T^{\dagger}=-T

, on a complex inner-product space. Prove that every eigenvalue of

T

is purely imaginary, and that eigenvectors belonging to distinct eigenvalues are orthogonal. (b) A critically damped galvanometer needle obeys

\ddot x + 2\omega_0\dot x + \omega_0^2 x=0

with

\omega_0 = 3\,\mathrm{rad\,s^{-1}}

. Reduce this to

\dot{\mathbf{x}}=\mathsf{A}\mathbf{x}

, find the Jordan block for

\mathsf{A}

, and use the truncated series for

e^{t\mathsf{A}}

to find

x(t)

for release from

x(0)=5\,\mathrm{cm}

at rest. Evaluate

x(0.5\,\mathrm{s})

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